@MISC{Culik96anaperiodic, author = {Karel Culik and II and Jarkko Kari}, title = {An aperiodic set of Wang cubes}, year = {1996} }

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Abstract

We introduce Wang cubes with colored faces that are a generalization of Wang tiles with colored edges. We show that there exists an aperiodic set of 21 Wang cubes, that is, a set for which there exists a tiling of the whole space with matching unit cubes but there exists no periodic tiling. We use the aperiodic set of 13 Wang tiles recently obtained by the first author using the new method developed by the second. Our method can be used to construct an aperiodic set of n-dimensional cubes for any n 3. Key Words: discrete mathematics, automata theory, aperiodic tilings, Wang tiles, Wang cubes, sequential machines 1 Introduction Wang tiles are unit square tiles with colored edges. The tile whose left, right, top and bottom edges have colors l; r; t and b, respectively, is denoted by the 4-tuple (l; r; t; b). A tile set is a finite set of Wang tiles. Tilings of the infinite Euclidean plane are considered using arbitrarily many copies of the tiles in the given tile set. The tiles are p...